Basic characteristics of uniaxial extension rheology: Comparing monodisperse and bidisperse polymer melts

Transcription

1 Basic characteristics of uniaxial extension rheology: Comparing monodisperse and bidisperse polymer melts Yangyang Wang, Shiwang Cheng, and Shi-Qing Wang a) Department of Polymer Science and Maurice Morton Institute of Polymer Science, The University of Akron, Akron, Ohio Synopsis We have carried out continuous and step uniaxial extension experiments on monodisperse and bidisperse styrene-butadiene random copolymers (SBR) to demonstrate that their nonlinear rheological behavior can be understood in terms of yielding through disintegration of the chain entanglement network and rubber-like rupture via non-gaussian chain stretching leading to scission. In continuous extension, the sample with bidisperse molecular weight distribution showed greater resistance, due to double-networking, against the yielding-initiated failure. An introduction of 20 % high molecular weight (10 6 g/mol) SBR to a SBR matrix ( g/mol) could postpone the onset of non-uniform extension by as much as two Hencky strain units. In step extension, the bidisperse blends were also found to be more resistant to elastic breakup than the monodisperse matrix SBR. Rupture in both monodisperse and bidisperse SBR samples occurs when the finite chain extensibility is reached at sufficiently high rates. It is important to point out here that these results along with the concept of yielding allow us to clarify the concept of strain hardening in extensional rheology of entangled polymers for the first time. a) Electronic mail: swang@uakron.edu 1

2 I. INTRODUCTION Extensional rheological behavior of entangled polymer melts has been studied for several decades. A clear understanding of failure behavior and material cohesive strength in extensional deformation is important for material designs. In the 1970s and 80s, Vinogradov and coworkers carried out extensive studies on the failure behavior of monodisperse entangled polymer melts in uniaxial extension [Vinogradov (1975); Vinogradov et al. (1975a, 1975b); Vinogradov (1977); Vinogradov and Malkin (1980); Malkin and Vinogradov (1985)]. During the same period, the failure behavior of various commercial polydisperse polymers were also studied by several teams [Takaki and Bogue (1975); Ide and White (1977, 1978); Pearson and Connelly (1982)]. It has been realized that polydispersity in the molecular weight distribution significantly affects the failure behavior in uniaxial extension [Takaki and Bogue (1975)]. Since most of the commercial synthetic polymers are polydisperse, a general understanding of the influence of polydispersity on failure behavior is of great industrial value. A first step towards a better understanding of the molecular weight distribution effect is to study bimodal blends where the behavior and dynamics of each individual component can be readily established. Most of the previous investigations of such systems [Minegishi et al. (2001); Ye et al. (2003); Nielsen et al. (2006)] focused on the "strain-hardening" characteristic during uniform extension, leaving the failure phenomena largely unexplored. Our recent studies on a series of monodisperse linear SBR melts [Wang et al. (2007b); Wang and Wang (2008)] have suggested that certain failure behavior of highly entangled polymers in rapid uniaxial extension is analogous to the shear inhomogeneity revealed by particle-tracking velocimetry [Tapadia and Wang (2006); Wang (2007)], and can be understood in terms of the disintegration of the chain entanglement network. Specifically, in both startup shear and extension, entangled polymers exhibit the same scaling characteristics associated with the yield point in the elastic deformation regime [Wang et al. (2007b)], which is the point where the shear stress and engineering stress peak. Beyond the yield point, structural inhomogeneity develops in the form of 2

3 non-uniform spatial distribution of chain entanglement. The purpose of this study is to demonstrate that the notion of yielding can be readily applied to explain some basic rheological characteristics of entangled bimodal blends in uniaxial extension, including their failure behaviors. Moreover, this concept of yielding allows us to clarify when strain hardening really happens in uniaxial extension of entangled polymers. II. MATERIALS AND METHODS The bimodal blends in the current investigation were made from three of near-monodisperse linear styrene-butadiene random copolymers (SBR) provided by Dr. Xiaorong Wang at the Bridgestone Americas Center for Research and Technology. The molecular characteristics of three SBR melts can be derived from small amplitude oscillatory shear measurements using a Physica MCR 301 rotational rheometer equipped with 25mm parallel plates. Two samples involve 20 wt. % of SBR1M in two different "matrices" of SBR240K and SBR70K respectively, and are labeled as 240K/1M (80:20) and 70K/1M (80:20) respectively. A third mixture has the composition of SBR240K and SBR1M given by 240K/1M (90:10). The small amplitude oscillatory measurements of the bimodal blends are shown in Fig. 1, from which some basic information is obtained as shown in Table I, where the equilibrium melt shear modulus G eq is determined as the value of G' at the frequency, at which G" shows a minimum. The Rouse relaxation time τ R of the sample was estimated as τ/3z, according to the tube model [Doi and Edwards (1986)]. Due to their slight microstructure and polydispersity differences, the terminal relaxation time τ of these samples does not scale with the molecular weight M as: τ ~ M 3.4. TABLE I. The Molecular Characteristics of SBR Melts Sample M n (kg/mol) M w /M n G eq (MPa) Z τ (s) τ R (s) SBR70K SBR240K SBR1M

4 Storage/loss modulus (Pa) K/1M (80:20) G' 240K/1M (80:20) G" 70K/1M (80:20) G' 70K/1M (80:20) G" Angular frequency (rad/s) Figure 1 Small amplitude oscillatory shear measurements of the bimodal blends at room temperature. The storage modulus and the loss modulus are represented by the open and filled symbols respectively. The uniaxial extension experiments were carried out using a first generation SER fixture [Sentmanat (2004); Sentmanat et al. (2005)] mounted on the Physica MCR 301 rotational rheometer. The failure behaviors of pure melts and blends were investigated in two different types of testing. One was the failure during startup continuous uniaxial extension at a constant Hencky strain rate. The other was the failure during step extension where the sample was suddenly subjected to a certain amount of strain. The specimen failure is video-recorded to allow post-experiment analyses. III. RESULTS A. Continuous Extension Two types of failure mode were observed during startup continuous extension of the pure SBR melts. During low-rate extension of SBR240K and SBR1M melts the sample breakage is found to initiate from non-uniform extension. At sufficiently high rates rubber-like rupture was found for SBR1M when the sample broke sharply and the original cross-sectional dimensions returned after rupture, indicating that no part of the specimen suffered much irrecoverable deformation (i.e., yielding or flow) Fig. 2(a) and Fig. 3 present the stress-strain curves of SBR240K and SBR1M melts at various rates. The experiments ending in rupture are represented by open symbols. In filled symbols we see that the engineering stress engr always exhibits a maximum such that ( engr ) max is linearly proportional to max. This characteristic has been reported 4

5 before and suggested to be a signature of yielding [Wang et al. (2007b), Wang and Wang (2008)], beyond which chains mutually slide past one another [Wang and Wang (2009)]. The engineering stress is also presented as a function of time in Fig. 2(b) for SBR240K. It is easy to see that the previously reported scaling behavior in the elastic deformation regime σ engr ~ t 1/2 is valid for SBR240K as well [Wang et al. (2007b)]. The end of each stress-strain curve corresponds to the onset of non-uniform extension by visual inspection, i.e., by examining the recorded images of the stretched specimen. On the other hand, at high rates rupture of SBR1M truncates the monotonic rise in the engineering stress engr. In other words, engr never had a chance to decline before rupture. Note that the data at 1 s -1 in Fig. 3 approaches the borderline between yielding and rupture. engr (MPa) (a) max SBR240K s s s s s s s Hencky strain, engr (MPa) (b) 15 s s s s s -1 SBR240K s s Time (s) Figure 2 Engineering stress as a function of (a) Hencky strain and (b) time for SBR240K at various strain rates. The straight dashed line in (a) provides an indication of how the engineering stress increases more weakly than linearly with the Hencky strain. engr (MPa) 10 SBR1M 7.5 neo-hookean 6.0 s s s s s s Hencky strain Figure 3 Engineering stress as a function of Hencky strain for SBR1M at various strain rates. The stretching, ending in yielding-initiated failure and rupture, is represented by filled and open symbols respectively. The dotted curve is the neo-hookean formula of Eq. (1) with G eq = 0.85 MPa from Table 1, showing exponential growth with the Hencky strain. 5

6 The engineering stress-strain curves for startup continuous stretching of bimodal SBR blends are shown in Fig. 4, Fig. 5 and Fig. 6 respectively. The shape of the engineering stress-strain curve for bimodal SBR blends is qualitatively different from that of the pure SBR melts. At various rates, all three blends, i.e., 240K/1M (90:10), 240K/1M (80:20) and 70K/1M (80:20), exhibited an engineering stress-strain curve containing two maxima and only showed non-uniform extension beyond the second maximum. At sufficiently high rates, the uniaxial extension of 240K/1M (80:20) and 70K/1M (80:20) was terminated abruptly without displaying a second maximum in the engineering stress when the specimens ruptured. The rate dependence of strains at yielding-initiated failure and rupture for SBR240K, 240K/1M (90:10), and 240K/1M (80:20) is shown in Fig. 7. The onset of both failures in the bimodal blends is significantly postponed at the highest four rates relative to those of the pure components K/1M (90:10) 1.5 engr (MPa) s s s s s s s Hencky strain Figure 4 Engineering stress as a function of Hencky strain at various strain rates for the 240K/1M (90:10) blend. 3 engr (MPa) K/1M (80:20) 15 s s s s s s s s Hencky strain Figure 5 Engineering stress as a function of Hencky strain at various strain rates for the 240K/1M (80:20) blend. The stretching, yielding-initiated failure and rupture, is represented by filled and open symbols respectively. 6

7 Failure Hencky strain K/1M (80:20) 10 s s -1 engr (MPa) 1 15 s s s s s Hencky strain Figure 6 Engineering stress as a function of Hencky strain at various strain rates for the 70K/1M (80:20) blend. The stretching, yielding-initiated failure and rupture, is represented by filled and open symbols respectively. 10 SBR240K 240K/1M (90:10) 240K/1M (80:20) 1 Elastic rupture (80:20) Elastic rupture (0:100) Strain rate (1/s) Figure 7 Failure Hencky strain as a function of applied strain rate for SBR(240K) and the two bimodal blends. The solid symbols represent ductile failure through yielding. The dashed lines show the borderline between viscoelastic and elastic regimes for the two blends. The open symbols denote strains at rupture for the pure SBR1M (triangles) and the 240K/1M (80:20) blend (diamond). B. Step Extension The engineering stress as a function of time during and after step extension of the pure SBR1M melt is first presented in Fig. 8 at three different amplitudes all high enough to produce elastic yielding, i.e., failure during relaxation. The induction times, before which the specimens appear intact, are all longer than the Rouse relaxation time R of 7.2 s. After this period, one portion of the specimen started to shrink in its dimensions, leading to the sample breakage. Fig. 9(a) shows the elastic breakup behavior of not only the pure SBR240K in solid symbols but also the bimodal SBR blends of 240K/1M 7

8 (90:10). Here the critical Hencky strain for the breakup is just beyond 0.6 for the pure SBR240K, which is slightly lower than the vinyl-rich SBR melts previously reported [Wang et al. (2007b)]. The critical strain for SBR 240K/1M (90:10) is apparently the same as the pure SBR240K melt, although the induction time to break is markedly longer. 10 SBR1M Applied rate: 15 s -1 engr (MPa) strain Time (s) Figure 8 Engineering stress as a function of time in step relaxation experiments for the SBR240K melt and the 240K/1M (90:10) blend. 10 (a) Applied rate: 15 s -1 engr (MPa) strain K 240K/1M (90:10) Time (s) 2 1 (b) 240K/1M (80:20) Applied rate: 15 s -1 engr (MPa) 0.1 Strain Elastic breakup No breakup Time (s) Figure 9 Engineering stress as a function of time in step relaxation experiments for (a) the SBR240K melt and the 240K/1M (90:10) blend, and (b) 240K/1M (80:20) blend. In contrast, the critical Hencky strain c for SBR 240K/1M (80:20) shifted to c = 0.9, significantly higher than c ~ 0.6 for the pure SBR240K. Thus, the bimodal blends are 8

9 also more resistant to breakup after a step extension than the corresponding pure monodisperse matrix. IV. DISCUSSION A. Continuous Extension A.1 Yielding and rupture of Monodisperse Melts Yielding and non-uniform extension During startup deformation at rates involving Weissenberg number Wi =, after the initial elastic deformation, the entanglement network is forced to disintegrate when an imbalance emerges between the elastic retraction force and intermolecular locking force [Wang et al. (2007a), Wang and Wang (2009)]. In other words, the tensile force represented by engr is expected to display a maximum. The present study confirms that pure monodisperse melts exhibit yielding in constant-rate uniaxial extension. The yielding is signified by the emergence of a maximum in the engineering stress as shown in Fig. 2(a). Consistent with the reported scaling behavior of yielding in simple shear [Boukany et al. (2009a)], the solid symbols of Fig. 2(a) and Fig. 3 also show a shift of the yielding strain (corresponding to the engineering stress maximum) ε y to higher values upon increasing the applied rate. The broad maxima at lower rates in Fig. 3 are due to the large polydispersity in the molecular weight distribution of SBR1M. There is strong evidence that the observed sample necking is not due to an elastic Considère-type instability, advocated recently by Hassager and coworkers [Lyhne et al. (2009); Hassager et al. (2010)]: When set to a stress-free state after the engineering stress maximum, the specimen cannot return to its original dimensions [Wang and Wang (2008)]. The plastic deformation and the corresponding decline in the engineering stress are a result of the entanglement network disintegration in presence of the continuing extension. Apparently, the eventual outcome is, not surprisingly, that one portion of the stretched specimen reaches a point of network disintegration and irreversible deformation before the rest of the specimen does, resulting in necking or lack of uniform extension. It is interesting to note from Fig. 2(a) that the specimen can extend uniformly well after the yield point for the group of data involving the higher rates 9

10 of 6.0 s -1 and above. This dividing line coincides with the boundary between the viscoelastic and elastic deformation regimes, given by the condition of the Rouse Weissenberg number Wi Rouse = τ R ~ 1 [Wang and Wang (2008)]. In the viscoelastic regime, the entanglement network could undergo irreversible deformation even before reaching the engineering stress maximum, so that the specimen is more ready to fall apart beyond the stress maximum. Conversely, in the elastic deformation regime, the network would be largely intact and fully recoverable up to the yield point (engineering stress maximum) [Wang and Wang (2008)]. As a consequence, the onset of structural collapse is delayed markedly beyond the yield point. We close this subsection by emphasizing that nearly monodisperse linear melts such as SBR240K only show significant strain softening in the rate range bounded by Wi Rouse < 10. Fig. 2(a) shows, consistent with previous analyses [Wang et al. (2007b), Wang and Wang (2008)], that engr initially grows with the Hencky strain = t linearly. This may be expected because even the neo-hookean model prescribes such a linear response at small extensions: engr = G( 1/ 2 ), which has a limiting form of engr ~ 3G for << 1 where the stretching ratio =exp( ) and G is the elastic shear modulus. engr ~ is simply the straight dashed line in Fig. 2(a). However, the actual engr soon bends downward in Fig. 2(a), which is a linear scale plot of engr versus Hencky strain. Thus, the engineering stress build-up is even weaker than the linear relation of Eq. (1) at higher strains. We have taken this behavior to imply strain softening. Although this strain softening may just reflect weakening of the entanglement network, we note that chain relaxation during stretching can also partially contribute to this apparent strain softening. Rubber-like Rupture For monodisperse entangled linear melts, the transition from yielding to rubber-like rupture typically takes place at a critical extensional rate corresponding to Rouse Weissenberg number Wi Rouse ~ 10 [Wang and Wang (2010a)]. This transition is what Malkin and Petrie (1997) referred to as the rubber-to-glass transition connecting regime 10

11 III and IV. However, our recent study [Wang and Wang (2010a)] has revealed that for highly entangled polymers, such a yielding-to-rupture transition is located in the middle of the rubbery plateau region, and does not involve any transformation to the glassy state. In other words, it would be misleading to associate the rupture with the term glass-like zone, as classified in the Malkin-Petrie master curve. The ductile failure arises from yielding of the entanglement network, during which linear chains mutually slide past one another [Wang et al. (2007a); Wang and Wang (2009)], whereas the origin of the rupture could have something to do with chain scission. At the very high rates, the mode of yielding, i.e., chain mutual sliding, is no longer available when sufficient tension in the chains can build up to cause chain scission. In other words, chain scission would occur before chains have reached the point of force imbalance [Wang and Wang (2009) to allow mutual sliding. This appears to occur in SBR1M at an extensional rate = 2.0 s -1 and above as shown in Fig. 3 where the tensile force (i.e., the engineering stress) rises monotonically. This rapid rise in the engineering stress at higher rates is plausibly due to non-gaussian stretching [Wang and Wang (2010a)]. In the explored rate range, SBR240K cannot be stretched fast enough at room temperature to produce the non-gaussian stretching that leads to the upward rise in tensile force and ends in rupture via chain scission. In passing, we note that some alternative theoretical explanation for specimen failure during uniaxial extension has been made in the literature [Joshi and Denn (2003, 2004)]. We do not elaborate on this theoretical study because it appears to have oversimplified the process leading to the yielding-initiated failure. A.2 Failure Behaviors of the Bimodal Blends Effect of the high molecular weight component on yielding We have seen that the incorporation of a small amount of SBR1M to SBR240K significantly delays the sample failure. For example, at = 6.0 s -1, the SBR240K melt would yield at 1.4 and eventually undergo ductile failure at 2.6. The presence of 10% or 20% 1M SBR postpones the onset of the ductile failure, as shown in Fig. 4 and Fig. 5, to much higher strains. A closer examination shows that there exists a critical 11

12 rate, beyond which the strength of the blends is greatly improved. As evident from Fig. 7, the failure behavior of the 240K/1M (90:10) blend is not so different until reaching c ~ 6.0 s -1. Similarly, c ~ 2.0 s -1 can be found for the 240K/1M (80:20) blend. The relaxation times of the two individual components in the 240K/1M blends are widely separated. As indicated in a preceding subsection, the Rouse relaxation times of 240K and 1M are different by a factor of 60, whereas the difference in their terminal relaxation times is even larger, approaching 330. The wide separation of relaxation times causes the high and the low molecular weight components to stay in different deformation regimes under a given applied strain rate. In other words, the incorporated SBR1M chains actually formed a second entanglement network, which is highly plausible given the huge separation of its relaxation time scale from that of the "matrix" SBR240K. Specifically, the number of entanglements per chain in such a second network can be estimated according to the empirical scaling law for the entanglement molecular weight: M e ( ) ~ M e, [Yang et al. (1999)]. At the volume fractions of 0.1 and 0.2, the second network involves respectively 32 and 73 entanglement points per SBR1M chain. Our previous study on the scaling characteristics of yielding [Wang and Wang (2009)] has revealed, as also demonstrated by Fig. 2(a), that the onset of yielding moves to higher strains with increasing Rouse Weissenberg Number Wi Rouse. Thus, at a given rate, the second entanglement network formed by the SBR1M would yield at a much higher strain. The blend could retain its integrity well beyond the first engineering stress maximum where the (first) matrix network has collapsed. This evidently indicates that the second entanglement network made of SBR1M did not give in until much higher strains. Since the second network involves much greater entanglement spacing, the non-gaussian stretching sets in at significantly higher strains around a Hencky strain of 3.0 as read from Fig. 5 [Wang and Wang (2010b)] than a Hencky strain of 2.0 read from Fig. 3 for the neat SBR1M. More discussion on non-gaussian stretching is deferred to IV.D.4. 12

13 Rupture With only 10% SBR1M, the first blend did not undergo rupture in the explored rate range as shown in Fig. 4. It appears that at these rates the second network associated with 10% SBR1M never got extended sufficiently to suffer chain scission. As a consequence, it yields in the form of mutual chain sliding. It is truly remarkable that the two blends of 240K/1M (80:20) and 70K/1M (80:20) containing only 20 % of SBR1M would undergo rupture in the range of Hencky rates where the pure "matrices" only show yielding-initiated non-uniform extension at the same rates. Apparently at these rates there is not sufficient intermolecular gripping force to produce full chain extension and enough chain tension to cause chain scission in either pure SBR240K or SBR70K. Thus, the 80% matrix chains only yield in the form of mutual chain sliding. On the other hand, the second network due to SMR1M can be fully stretched to reach the point of chain scission. Apparently, the 80% matrix chains at the point of rupture have reached such a fully disengaged state that the creation of the new surface during rupture costs no more energy than that estimated from the surface tension. In other words, the incorporation of 20% SBR1M as a second network provided so much structural stability that the matrix, i.e., the first network made of SBR240K or SBR70K, was able to disintegrate fully. The rupture occurred at much higher strains in the blends than in the neat SBR1M because at the volume fraction of 0.2, a strand (made of SBR1M) between two neighboring entanglement points of the second network is of a much longer chain length. Our recent study has shown that a higher stretching ratio is required to produce full chain extension, which appears to be a necessary condition for chain scission [Wang and Wang (2010b)]. The difference between the open diamond and triangles in Fig. 7 indicates rupture at rather different stretching ratios due to the difference in entanglement density. Finally, it is more than interesting to note that the blend of 240K/1M (80:20) barely shows rupture at = 15 s -1, whereas the blend with SBR70K as the matrix would undergo rupture at a rate as low as = 6 s -1. We know that at the same rate SBR70K yields at a lower strain than SBR240K because the yield strain decreases with lowering Wi Rouse, which is lower for SBR70K. Consequently, SBR70K can reach a fully 13

14 disengaged state at a lower strain. As suggested above, the disappearance of any entanglement networking in the matrix appears to be a prerequisite for rupture. Apparently at = 6 s -1, when SBR1M suffers chain scission, the matrix is also already in a state of full disengagement, allowing rupture to take place in 70K/1M (80:20). Conversely, in 240K/1M (80:20) that is being extended at = 6 s -1, even when a significant fraction of SBR1M chains have become fully extended, evidenced by the upturn in the engr vs. curve of Fig. 5, and reached the condition for chain scission, and even if these chains do suffer scission, the matrix apparently has not fully collapsed at these strains. The engineering stress engr could further build up until most SBR1M chains have undergo chain scission. Beyond this point, the matrix chains further mutually slide past one another until the system lose uniform structural support from chain entanglement. This appears to be what is happening in the blend involving the stronger first network made of SBR240K. B. Step Extension Recent studies [Wang et al. (2007b); Wang and Wang (2008)] on the relaxation behavior of monodisperse entangled linear polymers after step uniaxial extension have revealed that the entanglement network suffers cohesive breakdown at large strains. Here the cohesion refers to that due to chain entanglement [Wang et al. (2007a)]. Such elastic breakup is analogous to those disclosed by the particle-tracking velocimetric observations of entangled polymer solutions [Wang et al. (2006); Ravindranath and Wang (2007)] and melts [Boukany et al. (2009b)] in startup simple shear. The dynamics of such elastic yielding appear to depend on the molecular weight of the polymer melt and the amplitude of the step strain. For both uniaxial extension [Wang et al. (2007b); Wang and Wang (2008)] and simple shear [Boukany et al. (2009b)], the elastic yielding takes place faster with increasing step strain amplitude. For the same amplitude of step strain, a sample with higher molecular weight takes a longer time to lose the cohesive integrity [Boukany et al. (2009b)]. In fact, the elastic yielding appears to be related to the Rouse chain dynamics in the sense that the induction time for the breakdown scales with the longest Rouse relaxation time [Wang et al. (2007a), Boukany et al. (2009b)]. Because 14

15 the Rouse relaxation time of SBR1M is 60 times of SBR240K (note that this ratio is far greater than Z 2 ~ 25, due to the polydispersity of SBR1M), the incorporation of incorporation of SBR1M may delay the specimen breakup after step extension as shown in Fig. 9(a) for the 240K/1M (90:10) blend. Upon a step extension of Hencky strain of 0.75 and 0.9, the pure SBR240K undergoes elastic yielding that leads to the sample failure at 3.5 and 10 s respectively as shown by the filled symbols in Fig. 9(a). The blend (open symbols) fails at appreciably later times of 6.6 and 13 s respectively. It appears that the specimen would not undergo failure until the second network associated with the high molecular weight SBR1M also collapse on some longer time scale. Something more dramatic occurs in the second blend containing 20% SBR1M. Here the presence of the SBR1M can prevent the sample from undergoing the cohesive failure. For a step extension of Hencky strain = 0.9, the blend does not suffer failure. Perhaps before the second network made of SBR1M undergoes any disentanglement the first network has already recovered its entanglement structure after experiencing elastic yielding. At the higher strain of 1.2, the disintegration of the second network perhaps occurred so quickly that the first network had no chance to return to its entangled state to provide any adequate structural support. Consequently, the sample failure was observed at this higher strain. Fig. 8 indeed shows that the breakup of the pure SBR1M accelerates from an induction of 28 s for the step extension of =0.9 to 11s for = 1.2. C. Elastic Instability or Yielding Over the past a few decades, there have been extensive studies of entangled polymer melts and solutions in extensional deformation, because of their theoretical and practical importance. Among all these studies, significant efforts have been made toward understanding the nature of specimen failure. Vincent (1960) was the first to use the Considère criterion [Considère (1885)] to analyze the necking instability in elongation and cold flow of solid plastics (PE and unplasticized PVC). Such an analysis was subsequently extended to polymer melts in the rubbery state [Cogswell and Moore (1974); Pearson and Connelly (1982)] and polymer solutions [Hassager et al. (1998); Yao et al. (1998)]. Doi and Edwards (1979) also suggested an elastic necking instability in their 15

16 original tube theory based on the Considère reasoning. McKinley and Hassager (1999) applied the Doi-Edwards theory and pom-pom model [McLeish and Larson (1998)] to predict the critical Hencky strain for failure in linear and branched polymer melts during fast stretching K 15 s -1 engr (MPa) s s s s s s Hencky strain 3.0 s K/1M (90:10) Figure 10 Comparison of the engineering stress-strain curves for the SBR240K melt and the 240K/1M (90:10) blend. In our view, the emergence of a maximum in the engineering stress implies weakening of the underlying structure. Beyond the maximum, the material is no longer the original elastic body. In rigid solids such as metals studied originally by Considère (1885), when the measured force passes a maximum, non-uniform extension take place immediately due to the very limiting amount of extensibility. The debate in the literature about how to apply the Considère criterion has been around the issue of whether necking could occur immediately after the force maximum [Barroso et al. (2010); Petrie (2009); Joshi and Denn (2004b)]. For rubbery materials including entangled melts, the force maximum occurs at a significant stretching ratio, and non-uniform extension usually does not occur right after the force maximum. Fig. 2(a) shows that the maximum occurs for = 15 s -1 at a Hencky strain of = 1.8, and the failure strain is at 3.4. This behavior renders the application of the Considère criterion irrelevant. The force peak is the yield point when the sample is weakening in its resistance against further external deformation. The entanglement network takes some further extension before disintegration [Wang and Wang (2009)]. In other words, uniform extension could persist for a while until the network structure of the sample becomes inhomogeneous. The eventual "necking" and failure are anything but a mechanical 16

17 (elastic) instability in our judgment. So far no theory can describe quantitatively how such a localization of yielding takes place. Any continuum mechanical depiction based on a non-monotonic relation between the tensile force and degree of extension would have to first explain the microscopic physics responsible for the tensile force decline with increasing stretching. More specifically, the phenomenon of the specimen failure after yielding is a result of localized cohesive failure, having to do with how the initial elastic deformation turns into plastic deformation in an inhomogeneous manner. Introduction of 10 or 20 % SBR1M long chains into the SBR240K produces similar engineering stress versus Hencky strain characteristics as the pure SBR240K up to some significant strains well beyond the maxima, as shown in Fig. 10. For example, at = 6.0 s -1, both the pure SBR240K and the blend show nearly identical stress-strain curve till = 2.7. A continuum mechanical analysis based on such curves would have predicted the onset of necking instability, i.e., non-uniform extension, at the same strain for both the pure SBR240K and its blend with SBR1M. This is far from truth: the presence of only 10 % SBR1M long chains significantly extended the range of uniform stretching till = 4.0, well beyond the strain where the necking and failure took place in the pure SBR240K. We assert that this suppression of non-uniform extension by the second entanglement network is something well beyond any existing theoretical description based on continuum mechanical calculations. More recently, Hassager and coworkers also tried to explain the delayed failure after rapid uniaxial extension (i.e., the elastic yielding according to Wang et al. (2007b)) on the basis of the Doi-Edwards model and a Considère-type analysis [Lyhne et al. (2009); Hassager et al. (2010)]. Since the observed elastic-yielding-initiated failures after step extension only involved a level of extension well below the tensile force maxima that occur around = 2.0, it is rather difficult to understand that this failure discussed in Fig. 9(a)-(b) has something to do with the emergence of the non-monotonic engineering stress vs. strain curves shown in Fig. 2(a) and 5. 17

18 Stress (MPa) D. Where Is Strain Hardening Since strain hardening is a frequently invoked expression of extensional rheological behavior of polymer melts, we will discuss thoroughly this idea in the present context. In particular, we will review and comment on four ways in which the concept of "strain hardening" may get used to describe uniaxial extension of polymers K/1M (80:20) neo-hookean Strain rate (s -1 ) Hencky strain Figure 11 Cauchy stress as a function of Hencky strain for the 240K/1M (80:20) blend. The dotted line represents the stress-strain curve from the neo-hookean model of Eq. (1) with G eq = 0.82 MPa from Table 1. D.1 Strain hardening with stress-strain curve above neo-hookean line One of the earlier references to strain hardening came from high extension of natural rubbers when the stress-strain curve appears above the prediction of the classical rubber elastic theory, or the neo-hookean model for a network of Gaussian chains [Treloar (1944)]. This strain hardening at high stretching ratios is partially due to the finite chain extensibility, i.e., the system has reached the point of non-gaussian stretching. Moreover, the strain-induced crystallization in natural rubber can also enhance strain hardening [Smith et al. (1964)]. However, entangled melts that do not undergo strain-induced crystallization, such as the present SBR melts and blends, only appear to show stress-strain curves beneath the neo-hookean model prediction as shown in Fig. 3 for the SBR1M and Fig. 11 for the 240K/1M (80:20) blend. Entangled polymer melts, in absence of chemical cross-linking, apparently always suffer so much loss of entanglements during extension that the overall stress-strain curve could not rise above the neo-hookean line even when non-gaussian stretching takes place. In other words, a 18

19 significant fraction of entangled points in the network reach the point of force imbalance and become ineffective to bear the load during continuous extension of entangled melts, which does not occur in the neo-hookean model. More discussion on this point is given in the following IV.D.4 D.2 Monotonic increase of stress in startup extension: false strain hardening At high Weissenberg numbers, startup deformation is known to produce stress overshoot in simple shear, which has recently been suggested to indicate the onset of yielding [Wang et al. (2007a), Wang and Wang (2009)]. Beyond the yield point, the shear stress or transient viscosity decreases because the elastic deformation ceases and disintegration of the entanglement network leads to reduced elastic resistance. In startup uniaxial extension, the so called true stress E, i.e., the product of the tensile force and stretching ratio = exp( t), would usually only monotonically grow with continuing extension until the point of non-uniform extension. In other words, in uniaxial extension, the extensional (Cauchy) stress E does not exhibit overshoot, and thus the transient viscosity E = / at a given Hencky strain rate only monotonically rises. This contrast between simple shear and uniaxial extension has caused Münstedt and Kurzbeck (1998) to declare that "The viscosity increase as a function of time or strain, respectively, is called strain hardening. It is a special feature of elongational deformation of polymer melts." In our view, this difference between simple shear and uniaxial extension should be looked at in a different light. Upon startup deformation the transient shear viscosity also initially increases with time, which is just an indication that the elastic deformation is dominant at the beginning of startup shear. The elastic deformation cannot ensue indefinitely, and yielding must take place, leading to the observed decrease of the shear stress as well as viscosity over time. In contrast, during startup continuous extension at Hencky rate the dominant reason for the continuous rise in E is the simple geometric factor of the exponentially shrinking cross-sectional area A(t): = F/A(t) = engr = engr (t) exp( t), (2) where F is the total tensile force, and 19

20 A(t)=A 0 exp( t) (3) is the time-dependent cross-sectional area with A 0 being the original area. Even if yielding occurs, i.e., engr (t) starts to decline, and E would still only increase with time until the point of non-uniform extension and specimen failure, provided that the decline of engr due to yielding is not as fast as the exponential areal shrinkage, which is often the case. Thus, the continuous increase of E and E does not mean that the uniaxial extension has not experienced the same yielding as seen in simple shear. In other words, such increases do not mean that the sample s entanglement structure is not deteriorating and becoming less resistant to the continuous extension. Actually, strain softening not hardening has taken place as long as the tensile force is declining with continuing extension. As far as we can tell, the point of the entanglement network weakening occurs at the engineering stress maximum and cannot be discerned readily from such quantities as the Cauchy stress and extensional viscosity. D.3 Upward deviation of transient viscosity from zero-rate limit There is yet another way in which strain hardening is referred to in uniaxial extensions of polymer melts. It refers to a specific phenomenon observed during startup uniaxial extension when the transient elongational viscosity shows upward deviation from the limiting zero-rate-viscosity vs. time curve, in contrast to the phenomenon in simple shear where the transient shear viscosity function is always below the zero-shear viscosity function. This is perhaps the most widely recognized signature for strain hardening. The phrase "strain hardening" might have first been used by Meissner (1975) in describing the uniaxial extensional behavior of low-density polyethylenes (LDPE) [Meissner (1971, 1975); Laun and Münstedt (1978)]. Linear melts with bidispersity in molecular weight distribution [Münstedt (1980); Koyama (1991); Münstedt and Kurzbeck (1998); Minegishi et al. (2001); Wagner et al. (2005); Nielsen et al. (2006)] and even monodisperse melts stretched at high enough rates [Wang and Wang (2008), and figures below] could also produce this upward deviation. Since LDPE shows the most pronounced upward trend, long chain branching has been thought 20

21 to play some peculiar role in producing this "strain hardening" [McLeish (2008); van Ruynbeke et al (2010)]. In our view, this noted difference between simple shear and uniaxial extension is perhaps superficial rather than fundamental for entangled polymer melts. Entangled polymer melts have been found to exhibit only strain softening in simple shear as a consequence of yielding during startup deformation. The maximum shear stress max has been found to grow with the applied rate more weakly than linearly [Boukany et al. (2009a)] so that the peak transient shear viscosity + max= max / can only decrease with. Moreover, the steady shear viscosity is always lower than the zero-shear steady-state viscosity: In the zero-rate limit, the entanglement network is intact, and it is the Brownian diffusion that brings the chains past one another. There is maximum viscous resistance to terminal flow because the equilibrium state of chain entanglement is preserved. In many cases, uniaxial extension is different in appearance only because the cross-sectional area A(t) of the sample keeps shrinking exponentially with time at a given Hencky strain rate as shown in Eq. (3). If one insists on representing the transient elastic response in terms of the Cauchy extensional stress of Eq. (2) and the transient extensional viscosity E (t) = / = engr (t)exp( t)/, (4) then the continuous increase of and E with time largely originates from the exponential factor associated with the shrinking cross-sectional area. Let consider the extension in the beginning in the sense that t < In the zero-rate limit, i.e., when Wi, we can estimate Eq. (4) by employing Eq. (1). When engr ~ E and exp( t) = 1, Eq. (4) turns into E (t) 0 = Et, for t <<1, (5) where 0 denotes the zero-rate limit and E = 3G. At very high rates, i.e., Wi >>1, E (t) is also given by Eq. (5) at short times. But at longer times, the exponential factor exp( 21

22 t) in Eq. (4) is a rapidly rising function of time. This causes Eq. (4) to grow above Eq. E (5) by the same exponential factor had engr (t) continued to rise linearly with the Hencky strain = t as shown by the dashed line in Fig. 2(a). In reality, the sample yields eventually. Thus, before the yield point when engr has not declined, the exponential factor in Eq. (4) kicks in to produce a higher E than the zero-rate curve. In other words, there is a small window of extension where the transient viscosity in Eq. (4) ticks upward as shown in Fig. 12. This upward deviation is largely due to the geometrical shrinkage of the transverse dimensions and is not true strain hardening. The less monodisperse sample of SBR1M can extend more before non-uniform extension terminates the experiments. In Fig. 3, although engr only grows by 40 % at = 1.0 s -1 from = 3.0 to the point of failure, Fig. 13(a) shows a much stronger rise in E (t) because of the exponential decreasing function A(t) of Eq. (3) in the dominator of Eq. (2) for the Cauchy stress E. As a consequence, the data rise significantly above the zero-rate envelope. Actually, whenever significant extension occurs without sample failure, there would be a great contribution to E from the cross-sectional areal shrinkage that has little to do with strain hardening. Therefore, from now on we shall call this upward deviation of the transient elongational viscosity from its zero-rate curve "pseudo strain hardening" SBR240K 3 *(1/ ) + (MPa.s) Strain rate (s -1 ) Time (s) Fig. 12 Transient extensional viscosity as a function of time for the SBR240K melt, where the "linear response" data given by η + E = 3 η*(1/ω) from the small amplitude oscillatory shear measurements are also presented as the reference [Gleissle (1980)]. 22

23 Long chain branching (LCB) in entangled melts delays the onset of non-uniform extension. For example, LCB in LDPE allows such materials to display a very shallow maximum in the engineering stress. In other words, the engineering stress only decreases gradually beyond its maximum. This gives the geometrical exponential factor a large range of time or strain to boost the transient viscosity in Eq. (4) above the limiting zero-rate curve, and caused the phrase "strain hardening" to be invoked to differentiate the extensional rheological behavior of LDPE from that of other linear polymer melts such as high-density polyethylene [McLeish (2008)]. It is clear that LCB plays a critical role to prolong uniform extension relative to entangled melts made of linear chains. However, engr does typically decrease with increasing extension in LDPE. In other words, there is only evidence of yielding and strain softening and little sign of strain hardening. Actually, there is explicit evidence from birefringence measurements that LDPE does not suffer non-gaussian chain stretching during extension in the typically explored range of strain rates and temperatures [Koyama and Ishizuka (1989); Okamoto et al. (1998)]. Finally, we need to explain why most literature data on linear melts show little upward deviation from the limiting zero-rate curve, a fact that makes LDPE look somehow special. Most experiments on uniaxial extension of linear polymer melts have been conducted in the moderately high (rather than extremely high) rate regime, and few have been based on monodisperse samples. For example, up to = 3.0 s-1, corresponding to Wi = 100, the data in Fig. 12 hardly tilted above the limiting curve, and some unimpressive upward deviation shows up only at the higher rates. As indicated above, E (t) would deviate exponentially fast above the zero-rate viscosity function if the sample would maintain linear growth of engr with the Hencky strain as shown by the dashed line in Fig. 2(a) In reality, the sample yields. The deviation of the actual data in Fig. 2(a) from this linear growth engr ~ E, i.e., the dashed straight line increases sharply with time. Beyond the engineering stress maximum, the deviation actually increases approximately exponentially, cancelling the exponential factor of exp( t) in Eq. (4) associated with the area shrinkage. Thus, at these intermediate rates, one can hardly 23

24 see any upward deviation in linear melts that readily suffers yielding. It is the yielding and the resulting non-uniform extension that makes it impossible to collect data points at longer times. As a consequence, it has been impractical to obtain the extensional viscosity in the fully developed steady flow state during startup continuous extension as noted by Petrie (2006). But Petrie did not know why it is almost impossible to reach the flow state [Petrie (2008)], which was due to uniform yielding as explained by Wang and Wang (2008). D.4 A case of entanglement strain hardening : non-gaussian stretching The transition from elastic extension to flow, known as yielding in engineering terms, occurs during startup uniaxial extension in a wide range of rates when further overall elastic deformation of the entanglement network is no longer possible and the chains mutually slide past one another. At higher rates, chain extension could continue until a fraction of the chains in the sample approaches the finite chain extensibility limit. Chains at such a high stretching ratio appear stiffer and non-gaussian. This is non-obvious from a conventional plot like Fig. 13(a). The mechanical evidence of this non-gaussian stretching comes from further analysis of data such as those in Fig. 3. It is obvious that some level of yielding, i.e., mutual chain sliding, can and does occur before a fraction of the yield-surviving entanglement strands reaches the finite chain extensibility limit. Fig. 13(b) shows that when the shear modulus G in Eq. (1) is reduced from its equilibrium value of 0.85 MPa to MPa, the neo-hookean would emerge onto the data at the applied Hencky strain rate of 6.0 s -1 at a stretching ratio of ng = 8.5, where the subscript ng stands for non-gaussian. Beyond this turning point, the data deviate upward from the neo-hookean curve, which can be taken as a sign of non-gaussian stretching. This upward deviation is true strain hardening at the chain level, which we shall call "entanglement strain hardening" to differentiate from the strain hardening in vulcanized rubbers that produces a stress-strain curve above the neo-hookean limit as discussed in IV.D.1. We see this behavior as shown in the open symbols in Fig. 3 for the SBR1M melt and in Fig. 5 and 6 for the binary mixtures. The full chain extension leads to chain scission and rupture during startup continuous 24

25 extension. One key characteristic for this strain hardening is the emergence of the upturn seen in Fig. 3 in open symbols in the pure SBR1M, and more instructively in Fig. E E 13(a). + (MPa.s) (a) = 1 s -1 3 *(1/ ) SBR1M Time (s) engr (MPa) (b) 6.0 s -1 ng neo-hookean SBR1M Figure 13 (a) Transient extensional viscosity as a function of time for the SBR240K melt, where the "linear response" data given by η E + = 3 η*(1/ω) from the small amplitude oscillatory shear measurements are also presented as a reference [Gleissle (1980)]. (b) Engineering stress engr versus the stretching ratio at = 6.0 s -1, relative to a neo-hookean curve of Eq. (1) based on G = MPa = G eq /2, i.e, half of the equilibrium shear modulus, implying that half of the strands in the equilibrium entanglement network are lost at the stretching ratio ng ~ (MPa.s) (a) 240K/1M (80:20) 3 *(1/ ) Strain rate (s -1 ) Time (s) engr (MPa) (b) Strain rate (s -1 ) ng 70K/1M (80:20) neo-hookean Figure 14 (a) Transient extensional viscosity as a function of time for the 240K/1M (80:20) blend, where the "linear response" data given by η E + = 3 η*(1/ω) from the small amplitude oscillatory shear measurements are also presented as a reference [Gleissle (1980)]. (b) Engineering stress engr versus the stretching ratio at the various rates for the 70K/1M (80:20) blend, relative to a neo-hookean curve of Eq. (1) based on G = G eq 2.2 = MPa where G eq = 0.85 MPa and = 0.2, where ng ~ 23, far higher than that of 8.5 for the pure SBR1M in Fig. 13(b). This upturn also shows up strongly in Fig. 5 and 6 that depict the two blends with the 20% long chains of SBR1M. It occurs whenever the limit of finite chain extensibility is approached to cause non-gaussian stiffening of the entanglement network. 25